If $\displaystyle f$, $\displaystyle g$ are $\displaystyle \mu$-measurable functions in a measure space $\displaystyle (S, \Sigma, \mu)$ such that $\displaystyle |f|^2$ and $\displaystyle |g|^2$ are $\displaystyle \mu$-integrable, show that the product $\displaystyle fg$ is $\displaystyle \mu$-integrable.

My first instinct was to apply this theorem (which my textbook doesn't give a name):
A function $\displaystyle f$ is $\displaystyle \mu$-integrable iff $\displaystyle f $ is $\displaystyle \mu$-measurable and there exists an integrable dominant for $\displaystyle f$
But I can't see any way to get the integrable dominant, although the measurable part is easy. Most of the previous exam questions for this subject were very easy so I think I might be missing something quite basic.

Thanks for bothering to read up to here, even if you can't help.

Nov 15th 2009, 10:17 PM

Moo

Hello,

For me, the first theorem that comes in mind is Cauchy-Schwarz inequality (Nod)
and it works, considering that a function is integrable if its integral is finite.